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DSC1007BusinessAnalytics

TheGeometryofLinearProgramming

Na#onalUniversityofSingapore

Lecture8

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OriginofOptimizationModeling Optimization:What?When?Why? Optimizationprocess:

OptimizationModel:3majorcomponents Objective Decisionvariables Constraints

Standardformulationofoptimizationmodels Examples:

production,transportation,investment

2

PracticalProblem

OptimizationModel

Solution

Recap:LinearOptimization

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Howtovisualizealinearprogramgeometrically? Decisionvariables? Constraints? Objective?

HowtoNindtheoptimalsolutiongeometrically? Anexample:

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GeometryofLinearProgramming

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Eachdimensionrepresentsadecisionvariableann-dimensionalspace

Eachpointinthespacerepresentsaparticularsolution

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X1

X2

0

GeometryofLPDecisionVariables

(2, 3)3

2

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Constraint:x10 Inequalityconstraintx10deNinesahalfspace

5

X1

X2

X1 0

GeometryofLPConstraints

0

x1 = 0

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Constraint:x20 Inequalityconstraintx20deNinesahalfspace

6

X1

X2

0

X2 0

GeometryofLPConstraints

x2 = 0

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Constraint:2x1+x23 Inequalityconstraint2x1+x23deNinesahalfspace

7

X1

X2

0

2x1 + x23

GeometryofLPConstraints

2x2 +x2 = 3(0, 3)

(1.5, 0)

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Constraint:x1+2x23 Inequalityconstraintx1+2x23deNinesahalfspace

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X1

X2

0

x1 + 2x23

GeometryofLPConstraints

(3, 0)

(0, 1.5)

x2 +2x2 = 3

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TheintersectionofalltheconstraintsFeasibleRegion

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X1

X2

0

GeometryofLPConstraints

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FeasibleRegion:Thesetofalltheallowedsolutions;aregion(polygon)boundedbytheconstraints EachequalityconstraintdeNinesaline EachinequalityconstraintdeNinesahalf-space

ExtremePoints:Cornerpointsontheboundaryofthefeasibleregion.E.g.,(0,0),(1.5,0),(0,1.5),and(1,1)

Infeasibleproblem:Aproblemwithanemptyfeasibleregion Redundantconstraint:Addingorremovingtheconstraintdoes

notaffectthefeasibleregion

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X1

X2

0

GeometryofLPConstraints

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Objective:maximizex1+x2 Isoquant:Alineonwhichallpointshavethesameobjective

value;allpointsareequallygoodontheobjectivefunction.

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X1

X2

0

x1

+ x2

= 0

GeometryofLPObjective

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Isoquant:x1+x2=1

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X1

X2

0

x1

+ x2

= 1

GeometryofLPObjective

(1, 0)

(0, 1)

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Isoquant:x1+x2=1.5

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X1

X2

0

x1

+ x2

= 1.5

GeometryofLPObjective

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Isoquant:x1+x2=3

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X1

X2

0

x1

+ x2

= 3

GeometryofLPObjective

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OptimalSolution:Thebestfeasiblesolution Theorem:ForanyfeasibleLPwithaNiniteoptimalsolution,

thereexistsanoptimalsolutionthatisanextremepoint

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X1

X2

0

Maxx1

+ x2

OptimalSolution:

(1,1)

GeometryofLPOptimalSolution

x1 + x2= 2

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OptimalsolutionsmayOTbeunique

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X1

X2

0

Max 2x1

+ x2

GeometryofLPOptimalSolution

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OptimalsolutionmayOTbeNinite

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X1

X2

0

Maxx1

+ x2

X1 - 2X2 2

-X1 + 2X2 2

GeometryofLPOptimalSolution

(2, 0)

(0, 1)

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Binding(oractive)constraints:TheconstraintsthataresatisNiedatequalityattheoptimalsolution

AllequalityconstraintsarebindingbydeNinition Non-binding(orinactive)constraintsaresatisNiedat

strictinequalityattheoptimalsolution

Theinequalitylevel(=RHSLHS)isknownastheslack BindingconstraintshavezeroslackbydeNinition

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GeometryofLPOptimalSolution

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Activeconstraintsaretheonesthatpassthroughtheoptimalsolution. Inactiveconstraintsaretheonesthatdonotpassthroughtheoptimal

solution.

Whyarebindingconstraintsimportant?

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Constraints Binding? Slack

X1 + 2X2 3 Yes 0

2X1 + X2 3 Yes 0

X1 0 No 1

X2 0 No 1

GeometryofLPOptimalSolution

Optimal

(1, 1)

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Decisionvariables

Eachdecisionvariableformsadimension

ndecisionvariablesdeNineann-dimensionalspace Asolutionisapointinthespace

Constraints EachequalityconstraintdeNinesahyper-plane(lineina2-Dspace) EachinequalityconstraintdeNinesahalf-space AllconstraintscollectivelydeNinethefeasibleregion

Objective TheobjectivefunctiondeNinesisoquantsandadirectioninthespace

ToNindingtheoptimalsolution,pushalongthedirectiondeNinedbytheobjectiveuntilwereachtheboundaryofthefeasibleregion

Attheoptimalsolution,someconstraintsarebinding(oractive)whileothersarenot

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GeometryofLPSummary

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AinNinitenumberoffeasiblesolutions

ANinitenumberofextremepoints Howmany? (+ )

Anextremepointistheoptimalsolution

TheSimplexalgorithmdevelopedbyDantzig

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GeometryofLPBeneNits

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